Created the Abstract Algebra theorems and definitions cheat sheet

part-3/chapters/chapter-18/chapter-18.tex

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+\chapter{Divisibility in Integral Domains}
+\subimport{./}{irreducibles-primes.tex}
+\subimport{./}{unique-factorization-domains.tex}
+\subimport{./}{euclidean-domains.tex}

part-3/chapters/chapter-18/euclidean-domains.tex

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+\section{Euclidean Domains}
+
+\begin{definition}[Euclidean Domain (ED)]
+	An integral domain $D$ is called a \textit{Euclidean domain} if there is a function $d$ (called the \textit{measure}) from nonzero elements of $D$ to the nonnegative integers such that
+	\begin{enumerate}
+		\item $d(a) \leq d(ab)$ for all nonzero $a,b \in D$; and
+		\item if $a,b \in D,\ b \neq 0$, then there exist elements $q$ and $r$ in $D$ such that $a = bq + r$, where $r = 0$ or $d(r) < d(b)$.
+	\end{enumerate}
+\end{definition}
+
+\begin{theorem}[ED Implies PID]
+	Every Euclidean domain is a principal ideal domain.
+\end{theorem}
+
+\begin{corollary}[ED Implies UFD]
+	Every Euclidean domain is a unique factorization domain.
+\end{corollary}
+
+\begin{theorem}[$\mathbf{D}$ a UFD Implies $\mathbf{D[x]}$ a UFD]
+	If $D$ is a unique factorization domain, then $D[x]$ is a unique factorization domain.
+\end{theorem}

part-3/chapters/chapter-18/irreducibles-primes.tex

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+\section{Irreducibles, Primes}
+
+\begin{definition}[Associates, Irreducibles, Primes]
+	Elements $a$ and $b$ of an integral domain $D$ are called \textit{associates} if $a = ub$, where $u$ is a unit of $D$. A nonzero element $a$ of an integral domain $D$ is called an \textit{irreducible} if $a$ is not a unit and, whenever $b$, $c \in D$ with $a = bc$, then $b$ or $c$ is a unit. A nonzero element $a$ of an integral domain $D$ is called a \textit{prime} if $a$ is not a unit and $a\ \vert\ bc$ implies $a\ \vert\ b$ or $a\ \vert\ c$.
+\end{definition}
+
+\begin{theorem}[Prime Implies Irreducible]
+	In an integral domain, every prime in an irreducible.
+\end{theorem}
+
+\begin{theorem}[PID Implies Irreducible Equals Prime]
+	In a principal ideal domain, an element is an irreducible if and only if it is a prime.
+\end{theorem}

part-3/chapters/chapter-18/unique-factorization-domains.tex

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+\section{Unique Factorization Domains}
+
+\begin{definition}
+	An integral domain $D$ is a \textit{unique factorization domain} if
+	\begin{enumerate}
+		\item every nonzero element of $D$ that is not a unit can be written as a product of irreducibles of $D$; and
+		\item the factorization into irreducibles is unique up to associates and the order in which the factors appear.
+	\end{enumerate}
+\end{definition}
+
+\begin{lemma}[Ascending Chain Condition for a PID]
+	In a principal ideal domain, any stricly increasing chain of ideals $I_1 \subset I_2 \subset \dots$ must be finite in length.
+\end{lemma}
+
+\begin{theorem}[PID Implies UFD]
+	Every principal ideal domain is a unique factorization domain.
+\end{theorem}
+
+\begin{corollary}[$\mathbf{\F[x]}$ Is a UFD]
+	Let $\F$ be a field. Then $\F[x]$ is a unique factorization domain.
+\end{corollary}